Introduction to Nonlinear Dispersive Equations
The aim of this textbook is to introduce the theory of nonlinear dispersive equations to graduate students in a constructive way. The first three chapters are dedicated to preliminary material, such as Fourier transform, interpolation theory and Sobolev s
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Introduction to Nonlinear Dispersive Equations
ABC
Felipe Linares
Gustavo Ponce
Instituto Nacional de Matem´atica Pura e Aplicada (IMPA) Estrada Dona Castorina 110 Rio de Janeiro-RJ Brazil [email protected]
Department Mathematics University of California Santa Barbara College of Letters and Science Santa Barbara CA 93106 USA [email protected]
ISBN 978-0-387-84898-3 DOI 10.1007/978-0-387-84899-0
e-ISBN 978-0-387-84899-0
Library of Congress Control Number: 2008939461 Mathematics Subject Classification (2000): 37Lxx:37L50
c Springer Science+Business Media, LLC 2009 ° All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Cover design: SPi Publisher Services Printed on acid-free paper springer.com
Preface
The goal of this monograph is to present an introduction to a sampling of ideas and methods from the subject of nonlinear dispersive equations. This subject has been of great interest and has rapidly developed in the last few years. Here we will try to expose some aspects of the recent developments. The presentation is intended to be self-contained, but we will assume that the reader has knowledge of the material usually taught in courses of theory of one complex variable and integration theory. This monograph is the product of lecture notes used for mini-courses and graduate courses taught by the authors. The first version of the lecture notes were written by Gustavo Ponce with Wilfredo Urbina from the Universidad Central de Venezuela and designed to teach a mini-course at the Venezuelan School of Mathematics in M´erida, Venezuela, in 1990. A second version of those notes was presented by Gustavo Ponce at the Colombian School of Mathematics in Cali, Colombia in 1991. These notes comprise a part of the materials covered in the first six chapters of the present monograph. Most of the original notes were used to teach various graduate courses at IMPA and UNICAMP by Felipe Linares. During these lectures the previous versions were complemented with some new materials presented here. These notes were also used by Hebe Biagioni and Marcia Scialom from UNICAMP in their seminars and graduate courses. The idea to write the present monograph arose from the need for a more complete treatment of these topics for graduate students. Before going any further we would like first to give a notion of what a partial differential equation of dispersive type is. We
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